論文の公開元へStudies on invariants of strictly pseudoconvex CR manifolds
概要
A CR manifold is an odd-dimensional analog of a complex manifold. A typical example of a CR manifold is a real hypersurface in a complex manifold. In this thesis, we consider strictly pseudoconvex CR manifolds. In the seminal paper [Fef74], Fefferman has proved that two bounded strictly pseudoconvex domains in the complex Euclidean space are biholomorphic if and only if their boundaries, which are strictly pseudoconvex CR manifolds, are isomorphic as a CR manifold. Since then, there have been extensive researches on strictly pseudoconvex CR manifolds. The purpose of this thesis is to study geometric properties of both local and global invariants of strictly pseudoconvex CR manifolds.
In Chapter 1, we will consider Chern classes of closed strictly pseudoconvex CR manifolds. It is one of the most fundamental problems in CR geometry whether a given CR manifold can be realized as a real hypersurface in a complex manifold. There are many strictly pseudoconvex CR manifolds of dimension three with no such realizations; see [Ros65, Nir74, BE90b] for example. On the other hand, it is known that any closed connected strictly pseudoconvex CR manifold of dimension at least five can be realized as the boundary of a strictly pseudoconvex domain in a complex projective manifold [BdM75,HL75,Lem95]. This fact gives some restrictions to the topology of closed strictly pseudoconvex CR manifolds. For example, Bungart [Bun92] and Popescu-Pampu [PP08] have proved some vanishing results for the cup product on the cohomology with rational coefficients. By a similar method, we will obtain a constraint on Chern classes (Theorem 1.1.1). Through some examples, we will also show that our result is “sharp” in general (Propositions 1.3.1 and 1.3.2). This chapter is based on the paper [Tak18b].
In Chapter 2, we will deal with the existence of a pseudo-Einstein contact form. Recently, two global CR invariants have been introduced: the total Q-prime curvature and the Burns-Epstein invariant, which will be explained later. A pseudoEinstein contact form, first introduced by Lee [Lee88], is a contact form satisfying a weak Einstein condition, and is necessary for defining those invariants. Therefore, the following problem arises when we consider the variation of these invariants: “is the existence of a pseudo-Einstein contact form preserved under deformations of CR structures?” We will solve this problem affirmatively for deformations as a real hypersurface in a fixed complex manifold (Corollary 2.1.2). This result follows from the fact that there exists a flat Hermitian metric on the canonical bundle of a sufficiently small tubular neighborhood of a strictly pseudoconvex real hypersurface if it admits a pseudo-Einstein contact form (Theorem 2.1.1). This chapter corresponds to the paper [Tak18c].
In Chapter 3, we will study relations between CR invariants constructed via the ambient space and Sasakian η-Einstein manifolds. Before stating our results, we recall the ambient space in even-dimensional conformal geometry, which is closely related to CR geometry.
Let N be a smooth manifold of dimension 2m ≥ 2, and C be a conformal class of pseudo-Riemannian metrics of signature (p, q) with p + q = 2m. The metric bundle G is the principal R+-bundle over N whose fiber Gx at x ∈ N is given by
Gx = { gx ∈ S 2T ∗ x N g ∈ C } .
This G has the tautological symmetric two-tensor g0. The ambient space is the space Ge = G ×(−ϵ, ϵ) with a homogeneous pseudo-Riemannian metric ge of signature (p + 1, q + 1) such that ge|G×{0} = g0 and ge is asymptotically Ricci-flat on G × {0}. This space was first introduced by Fefferman and Graham [FG85]; see also the book [FG12] for details. By using the Laplacian of ge, we can obtain some conformal invariants of (N, C). For each integer 1 ≤ k ≤ m, Graham, Jenne, Mason, and Sparling [GJMS92] have defined a conformally invariant differential operator Pk whose principal part is the k-th power of the Laplacian, called the k-th GJMS operator. It is known that P1 agrees with the conformal Laplacian; in other words, the GJMS operators are higher order analogs of the conformal Laplacian. The Qcurvature is a smooth function on N determined for each representative of C, first introduced by Branson [Bra95]; see also [FG02, FH03]. This is not a conformal invariant, but its integral, the total Q-curvature, defines a global conformal invariant of (N, C) if N is closed. In the case of m = 1, the Q-curvature coincides with the Gauss curvature; that is, the Q-curvature is a generalization of the Gauss curvature in higher dimensional conformal geometry.
Assume that C contains an Einstein metric g. Gover [Gov06], and Fefferman and Graham [FG12] have proved that the k-th GJMS operator is decomposed into k factors, and each factor is the sum of the Laplacian and a constant determined only by m and the Einstein constant. They also have shown that the Q-curvature with respect to g is a constant depending only on m and the Einstein constant. Moreover, the variation of the total Q-curvature under deformations of conformal structures is well-understood: the first variation of the total Q-curvature at C is always zero [GH05], and the second variation is written in terms of the Lichnerowicz Laplacian [Mat13]; see also [GMS16].
Now we return the CR case. Let M be a strictly pseudoconvex CR manifold of dimension 2n + 1. Then we can construct a principal S 1 -bundle N over M with a conformal class C of Lorentzian metrics on N; these are determined only by the CR structure of M [BDS77, Lee86]. The space (Ge, ge) with respect to (N, C) is called the ambient space of M. In this thesis, however, we take a complex-geometric approach to the ambient space following [HMM17]; see Section 0.3. Similar to the conformal case, the ambient space gives some CR invariants. For (w, w′ ) ∈ R 2 with k = w + w ′ + n + 1 ∈ {1, . . . , n + 1}, Gover and Graham [GG05] have defined a CR invariant powers of the sub-Laplacian Pw,w′ , a CR counterpart of the GJMS operators. A CR analog of the Q-curvature is the Q-prime curvature, introduced by Case and Yang [CY13], and Hirachi [Hir14]; it is a smooth function on M determined for each pseudo-Einstein contact form. Marugame [Mar18] has proved that its integral, the total Q-prime curvature, defines a global CR invariant for a closed M.
The results in Chapter 3 are CR counterparts of those in the paragraph before the previous one. A Sasakian η-Einstein manifold is a strictly pseudoconvex CR manifold (S, T1,0S) of dimension 2n + 1 with a contact form η satisfying a strong Einstein condition; see Definition 0.6.3. For such (S, T1,0S) and η, we will prove that Pw,w′ is factored into k components, and each component is written in terms of the sub-Laplacian, the Reeb vector field, n and the Einstein constant (Theorem 3.1.1). We will also show that the Q-prime curvature with respect to η is a constant determined only by n and the Einstein constant (Theorem 3.1.3). Note that the same results have been obtained independently by Case and Gover [CG17] in a different way. We will also consider the variation of the total Q-prime curvature at S under deformations as a real hypersurface. It will be proved that the first variation of the total Q-prime curvature at S must be zero (Proposition 3.1.4). By applying some spectral results, we will also show that, if n = 1 or the Einstein constant is non-negative, then the second variation is non-positive, and equal to zero if and only if the deformation is “infinitesimally trivial” (Theorem 3.1.5). Through some examples, it will be seen that these results do not hold if n ≥ 2 and the Einstein constant is negative (Theorem 3.1.6). This chapter is based on the paper [Tak18a]
In Chapter 4, we will compute another global CR invariant, the Burns-Epstein invariant, for the tubes associated with polarized Kähler-Einstein manifolds, which are typical examples of Sasakian η-Einstein manifolds. Burns and Epstein [BE90a] have introduced a global CR invariant for the boundaries of bounded strictly pseudoconvex domains in C n+1 as the boundary term of the renormalized Gauss-BonnetChern formula. Marugame [Mar16] has simplified their construction and generalized it to a global CR invariant for closed strictly pseudoconvex CR manifolds admitting a pseudo-Einstein contact form, which we call the Burns-Epstein invariant in this thesis. We will give a formula of this invariant for the tubes associated with polarized Kähler-Einstein manifolds in terms of characteristic numbers (Theorem 4.1.1). As an application, we will show that there is no proportional relationship between the Burns-Epstein invariant and the total Q-prime curvature in dimension at least five (Theorem 4.1.2). Note that, in dimension three, the Burns-Epstein invariant coincides with the total Q-prime curvature up to a universal constant.
